Physical systems often exhibit symmetry: we have already remarked on the symmetries of spanwise translation and reflection in boundary layers and shear layers and of rotations in circular jets. One could cite many more such cases. Of course, symmetric systems do not always, or even typically, exhibit symmetric behavior, and the study of spontaneous symmetry breaking is an important field in physics. These physical phenomena have their analogs in dynamical systems and in particular in ODEs, as we describe in this chapter.

The theory of symmetric dynamical systems and their bifurcations relies heavily on group theory and especially the notions of invariant functions and equivariant vector fields. The major references are the two volumes by Golubitsky and Schaeffer [134] and Golubitsky et al. [136]. In this chapter, as in the last, we attempt to sketch relevant parts of the theory using simple examples and without undue reliance on abstract mathematical ideas.